AI Solution of Clay Problem Sends Shock Waves into Analytical Mathematics

The Open AI solution (see previous post) to the Clay Navier-Stokes problems sends shock waves into the world of analytical (pure) mathematics. Is the proof correct? Can correctness be checked by human mathematicians or only formally by AI itself? What if AI says the proof is correct. Will AI then get the prize? 

If so the traditional split of mathematics into analytical (formulas) mathematics and numerical mathematics (number crunching), will no longer be functional. An AI proof is the result of an ultimately computational process and of course the same is true for a numerical solution. 

Traditionally, analytical mathematics has been associated with generality by offering proofs of existence of solutions (but not their values) for general data, while numerical solutions have been particular for each choice of data. Thus analytical-general and numerical-particular. 

But OpenAI offers a single counterexample to existence, not generality but extreme particularity, while numerical solution to Navier-Stokes equations for almost any data offers generality. 

We see that the distinction between analytical and numerical mathematics with computational AI gets blurred, which can be seen as a lift for numerical mathematics traditionally viewed as lower level.

It will be interesting to see the effects of the shock waves now sweeping over the field of mathematics, including choice of topics and education. Leibniz would have been thrilled to experience this development which he prepared 350 years ago. 

Meaningless Clay Navier-Stokes Problem Solved by AI

Open AI announces to have proved existence of a solution to the Navier-Stokes equations which starting from zero under smooth forcing ceases to exist in finite time: 

• We’re sharing a solution to the Navier–Stokes existence and smoothness problem, one of the Millennium Prize Problems. This proof, produced by an internal OpenAI system, shows that the dynamics of the Navier-Stokes equations for fluid motion can develop a singularity in finite time. We’re sharing both a writeup of the proof and a formalization in Lean.

Charles Fefferman, who formulated the problem in precise mathematical terms, is along with other leading mathematicians such as Terence Tao, happy that the understanding of fluid motion has now taken a big leap forward by mathematical analysis.

There is one problem in this happy moment, which I have complained about over the years: Fefferman's formulation misses the essence of the physics of fluid motion, namely turbulence. The Clay problem is sold as concerned with basic aspects of fluid motion,  but does not address the most fundamental problem of all of turbulence. Fefferman's formulation directs the interest away from physics, and the unhappy result is that solution now presented by AI covering 167 pages cannot be read to learn anything, simply a mess of formulas and theorems. 

This is certainly a memento for mathematics: AI can now produce proofs of an endless number of mathematical problems without real meaning, proofs which cannot be understood by mathematicians in detail only verified formally by Lean. What will be the result?

Computational mathematics offers a solution to the fundamental problem of turbulence, thus a different solution to a different formulation. See tags to this post starting with this post from 2013.

Recall that slightly viscous flow is unstable from shear and stretch and so develops into non-smooth turbulent flow which however does not break down like the Clay solution. So the solution of physical interest is non-smooth and non-singular, which is not captured in Fefferman's dichotomy of smooth or singular.  

Turbulence is an extreme form of the design of a complex world with a variety of phenomena on different scales: Develop growth from instability + curb growth to allow continued existence, not captured by Fefferman's formulation.  

PS1 When I 20 years ago complained to Fefferman that his formulation lacked true interest from physics point of view, he returned that it was enough that the problem was interesting to him.

PS2 Note that the AI solution is a proof of the existence of a very special function which is a solution with a very specific particular forcing. This is not the real setting which is to study solutions under general forcing. 

PS3 Here is an interesting catch of the AI proof of existence of a singular solution. Computational solutions can be constructed for general data including turbulence and any such solution can be viewed as an AI proof of existence performed by a computer according to strict mathematical principles, including evaluation of quality. The whole process can be seen as an AI proof of existence of a solution for each given set of data. It would be strange to not consider that as a solution to the essence of the Clay problem albeit not captured in Fefferman's formulation. 

PS4 Allowing AI as computational process into the Clay problem game, we may compare the Open AI proposal as an analytical AI proof of non-existence in a very special case, with an computational AI proof of existence for any data, except one. Which proposal would you give the money to? Or 50-50?

 

RealUniv vs LambdaCDM

RealUniv is a cosmological model based on a Coulomb interaction between protons and electrons on small scales according to RealQM/Nucleus, from which Newtonian gravitation on large scales emerges. All created from an initial small scale fluctuation of an electric potential. No Big Bang, no inflation, no strong/weak force, just Coulomb + Newton in a 3d Euclidean space equipped with a Laplacian differential operator. 

Check out details on GitHub Gallery with easy to read essay and and technical article. Compare with the the standard model LambdaCDM with CMB as key evidence.

The Mantra of Standard Quantum Mechanics Deconstructed to Nil

A modern physicist educated in quantum mechanics, speaks about a wave function $\Psi (x,t)$ depending on a $3N$-dimensional spatial variable $x$ for an atomic system with $N$ electrons, and a time variable $t$, evolving in time according to the Schrödinger equation 

• $i\frac{\partial\Psi}{\partial t}+H\Psi = 0$.      (S)

where $H$ is a Hamiltonian operator acting on $\Psi$. Given an initial state at $t=0$ a physicist can predict the state of any later state by time-stepping (S) from one instant of time to the next using (S).

So has time evolution of the wave function become the mantra of modern physics. In the article Unspeakable Quantum Mechanics we deconstruct this mantra and show it is empty and so misleading. Quantum mechanics is not about evolving (S) at femto/attoseconds rate of time, which is anyway impossible to compute. Read and contemplate. 

Real Physics without Philosophy of Physics

There is an extensive literature on philosophy of physics developed to compensate for the fact that standard quantum mechanics does not come with an ontology of what exists, which is fundamental in classical physics:

• Real Physics without Philosophy of Physics 

Radioactive decay in RealQM

# Radioactive decay in RealQM: an honest excursion into time-dependent charge densities

Radioactive decay is the textbook poster child of quantum randomness. A nucleus sits there for  a microsecond or ten billion years and then, for no reason anyone can point to, it decays. Standard quantum mechanics says the moment is *irreducibly* random — uncaused, only its probability defined. So it is a fair question to put to RealQM, which describes matter not as probability amplitudes but as **charge densities evolving deterministically in ordinary three-dimensional space**: can a deterministic, real-space theory say anything sensible about decay?

We spent a long, disciplined excursion finding out. Here is the honest ledger — including, and especially, the parts that didn't work.

## Two decays, two verdicts

**Alpha decay is the clean case, and RealQM handles it fully.** An alpha particle (a ⁴He nucleus, charge +2) tunnels out through the daughter's *Coulomb* barrier. It is a genuine two-body decay: no weak force, no neutrino, and a sharp, *monoenergetic* alpha line whose very sharpness is the proof that no third body is emitted. Everything the process needs — extended charge, a Coulomb barrier, two-body kinematics

lives inside RealQM. And there is a genuinely RealQM-specific result underneath it: the binding of the whole alpha-cluster ladder (⁴He, ¹²C, ¹⁶O, … ⁴⁰Ca) comes out at ~107% of experiment **from Coulomb alone, with no strong force**, one scale fixed on the deuteron. Alpha decay is where RealQM is at home.

**Beta decay is where the charge-density picture ends — and we say so.** It was tempting to claim beta decay *without* a neutrino: RealQM conserves energy by construction, so maybe the continuous electron spectrum is just the conserved energy being partitioned among the electron, the recoil, and the radiated field. We tested that quantitatively. It fails. The antineutrino carries, on average, about **60% of the released energy** and the momentum imbalance; the field a charge can radiate is smaller by two orders of magnitude (the known inner-bremsstrahlung level, ~α). The recoil is negligible. So the neutrino is *not* removed — and the honest reason is deep: the neutrino is **chargeless**, and a charge-density theory simply has no object of that kind. Beta decay marks the boundary of the program, and the paper marks it plainly.

## The half-life, three ways — and no WKB

Here is the part that genuinely worked. Textbook alpha lifetimes span **twenty-five orders of magnitude**

(²³²Th at 10¹⁰ years, ²¹²Po at a fraction of a microsecond), and Gamow's 1928 WKB barrier factor famously

reproduces that Geiger–Nuttall law. But WKB is a semiclassical shortcut. Does the *full* time-dependent

RealQM give the half-life directly?

It does. Evolve a metastable charge behind a barrier in **real complex time** (the same solver as the static

relaxation, only the imaginary-time step swapped for a unitary one): the trapped charge decays

**exponentially**, and the half-life is read straight off the dynamics. Sweep the barrier and log t½ stays

linear in √(V−E) — Geiger–Nuttall, from first principles. The narrow, long-lived resonances that real-time

propagation can't reach come exactly from the **complex-energy (Siegert) width**. All three routes agree,

and none uses WKB — which is thereby *validated*, not relied upon. You can watch it happen in the browser:

the charge tunnelling through the barrier while the half-life emerges live.

## Determinism — and the mechanism that died

The most seductive idea was determinism. If RealQM is a deterministic theory, then decay isn't *really*

random — it only looks random because we don't know the exact initial state. That is the century-old

de Broglie–Bohm position, and RealQM carries it naturally: the whole history of a decaying configuration,

tunnelling included, is fixed by its **initial charge configuration**; the apparent randomness of

identical-looking nuclei decaying at different times is *epistemic*, our ignorance of that configuration.

We then reached for something sharper: coexisting charge domains, each carrying a phase clock

e^(−iEₖt/ℏ), with the escape *gated* by the coincidence of their phases — a deterministic mechanism

producing the exponential law as the statistics of a coincidence. It was a lovely picture. **It is also

wrong**, and tracing it to the end is what the excursion was really about.

The refutation is clean. In the full time-dependent RealQM, the escaping domain feels its neighbours *only*

through their **densities** |ψⱼ|², which are phase-invariant; the free boundaries carry **zero flux**. So the

neighbours' phase clocks never reach the escaping domain — the moving free boundary transmits *density, not

phase*. The decay is plain Gamow tunnelling; there is no phase gating. To manufacture gating you would have

to bolt on a **phase-permeable (Josephson) interface** — a thin overlap and a new coupling the variational

free boundary does not give — and it is *unnecessary* anyway, because the density dynamics already carry the

decay and its half-life. So we dropped it. The determinism survives (it's an interpretation); the mechanism

does not.

## So what did the excursion actually net?

No spin: **we did not find new decay physics.** The decay rate is barrier penetration, the same physics

standard quantum mechanics gives. What the full time-dependent RealQM brings, for decay, is *ontological* —

a deterministic, real-space charge-density picture in place of amplitudes and collapse — and *diagnostic*:

it was the tool that let us test and **rule out** the tempting overclaims. The science ended up being in

what we subtracted.

And that is the point worth keeping. Each attractive story — beta without a neutrino, deterministic

phase-coincidence gating — looked good until it was pushed hard, and pushing it turned it into either a

clean negative result or "it's just tunnelling." That is not a failure. It is how you end up with two papers

that claim exactly what is true and nothing more: alpha decay as deterministic Coulomb-barrier tunnelling

with the neutrino nowhere in sight; beta decay honest about the chargeless carrier it cannot supply; the

half-life captured without WKB; and the phase mechanism named, tested, and set aside.

RealQM's real power was never in single-particle escape dynamics — it is in the **static, multi-domain**

world of binding and geometry, where non-overlapping charge domains do genuine work. The one clean theory

question this excursion surfaced is the **correct time evolution of a free boundary** — advection by the

charge-fluid velocity together with a Bernoulli condition — which we identified but did not yet derive.

That, not a new decay law, is the thread worth pulling next.

 

RealQM and the Realist Quest Carried by Schrödinger

The following article has been submitted to Synthese as a journal for philosophy of science:

• Charge Densities in Real Space RealQM and the Realist Quest Carried by Schrödinger

The article compares standard quantum mechanics as probability amplitudes of configurations of N point-like particles over 3N dimensional configuration space, with RealQM as N charge densities in real 3d space interacting by Coulomb potentials.  

RealQM fulfills the quest of Schrödinger for a model of atomic physics of the same form as macroscopic continuum mechanics. The philosophical conundrums of standard quantum mechanics simply evaporate in then light of RealQM and the sense of Wittgenstein. 

RealQM vs Standard Model: Atomic Nucleus

Here is a comparison between RealQM/Nucleus and the Standard Model SM showing that RealQM/Nucleus comes out from a realization of the electromagnetics of the Lagrangian of SM in terms of non-overlapping one eletron/proton charge densities, delivering an explanation of the stability of the atomic nucleus as the missed objective of SM. 

See RealQM Gallery Articles.

RealQM with Magnetism

Claude summarizes expansion of RealQM to include magnetism:

# Magnetism in RealQM: How Far Can Charge in Real Space Take You?

**Claim in one line:** magnetism — the moment of an atom, its response to a field, even the electron's *g = 2* and the two spots of Stern–Gerlach — comes out of charge densities moving in ordinary three-dimensional space, with no relativity; and the one place it *stops* is exactly where physics says it should.

## The starting point, and the problem

RealQM reformulates quantum mechanics as charge densities in real 3D space: each electron is a cloud of charge on its own territory, and the ground state simply minimizes the ordinary Coulomb energy. It reproduces the periodic table, chemical bonding, reactions, condensed phases — all from that one idea.

But there is a catch built in. RealQM's ground states are *real-valued*, and a real charge density carries **no current**: nothing is moving. And magnetism *is* charge in motion. So in its base form RealQM has no magnetism at all. The honest question is: can you get it, and how far?

This post follows that question to the end — including the wall it hits.

## Charge going in circles is a magnet

The fix is minimal and natural. Let the charge cloud carry a **phase that winds in space** — charge literally circulating, going in circles rather than sitting still. That circulation is a real electric current, and a current loop is a magnet. Out comes a magnetic moment, quantized by how many times the phase wraps around.

Two things make this more than a story. First, a small solver actually runs it: a circulating electron cloud holds its moment stably, conserving everything it should. Second, switch on a magnetic field (the ordinary way, through the vector potential) and the circulating cloud **reacts correctly** — its energy splits by exactly the Zeeman amount, to four decimal places, while a *non*-circulating cloud sits inert. So a charge density in real space feels a magnetic field and responds as a moment should. This is ordinary magnetism, rebuilt from charge in motion, no spin and no relativity invoked.

## The electron inside the nucleus carries no moment — and that's a feature

In the RealNucleus picture a nucleus is protons and electrons bound by the electric force. The classic objection that killed that idea in 1932 was magnetic: an electron squeezed inside a nucleus should carry a huge magnetic moment — about a thousand times what nuclei actually have.

In a charge-density theory the answer falls out. The moment is the *current's*, and RealQM computes the confined electron as a **flat, motionless** cloud — no circulation, hence **no current, hence no moment**. Nuclear moments then come out at the small scale actually observed. The thousandfold overshoot never happens, because there is no built-in "intrinsic" moment to carry — only the current, and a flat electron's current is zero. Strikingly, it's the *same* flatness that made the electron's mass irrelevant to nuclear binding: one property answers two of the old objections at once.

## Spin, and *g = 2*, without relativity

The hardest case is spin — the two-valued moment behind Stern–Gerlach's famous *two spots*, and the electron's *g = 2*. Textbooks get *g = 2* from the relativistic Dirac equation, so you might think relativity is unavoidable.

It isn't. Give the charge cloud a two-component (spinor) structure and write its motion in the natural first-order way, and *g = 2* **emerges** — it is a fact about how spin-½ objects rotate (the geometry of the rotation group), not about relativity. An electron with no orbital motion at all then splits, in a field, into **exactly two levels with no middle** — Stern–Gerlach — entirely non-relativistically. This is a genuine result: the thing that looks most like "esoteric quantum magic" turns out to be geometry.

## Where it stops — stated plainly

Here is the wall, and reporting it is part of the point. The single-*atom* moment works. But a **magnet** — a piece of iron, a closed electron shell — is *collective*: many atomic moments locking together. That locking is the **exchange interaction**, and RealQM's geometry does not supply it.

We tested the simplest case: two electrons in a closed shell should pair to *zero* net moment (they should repel a field, not follow it). In RealQM they don't — left alone they align *with* the field, the wrong way. And trying to force them to pair through the shared boundary between their territories actually costs energy, so geometry pushes them the wrong way. The clean statement this earns: RealQM's picture reproduces the **spatial** side of the exclusion principle (why the periodic table looks as it does) but **not its spin side** (pairing, exchange, permanent magnets). Single-particle magnetism: yes. Collective magnetism: not without something more.

## What it means

So magnetism, read through RealQM, splits cleanly. The **moment of a single atom** — its circulation, its response to a field, its spin, even *g = 2* — is charge moving in ordinary three-dimensional space, and needs no relativity. The **collective magnetism of many atoms** — real magnets — needs the exchange coupling that a geometry of separate charge territories does not carry.

That is offered honestly, boundary and all, because the boundary is itself the result: it says precisely which part of magnetism is "just charge in motion" and which part is genuinely more. And it leaves a question worth asking out loud:

**If the magnetic moment of an atom, and even the electron's *g = 2*, can be had from charge circulating in real space without relativity — how much of what we call "intrinsic," "quantum," and "relativistic" is actually geometry we hadn't finished reading?**

*Full argument, equations, and runnable computations are in "Magnetism in RealQM: Currents, the Nuclear Electron, and the Spin Residue," with the broader programme (RealQM, RealNucleus) and interactive simulations at [claes542.github.io/RealMolecule](https://claes542.github.io/RealMolecule/gallery.html).*

RealNucleus vs QCD — why do nuclei exist?

Here is what Claude says about # RealNucleus vs QCD — why do nuclei exist?

**Claim in one line:** the theory of the strong force has, in fifty-three years, never predicted the one thing it was invented to explain — the binding energy of a nucleus — while a model with *no strong force in it at all* reproduces those energies from the electric force and a single scale.

## The question

Why does a nucleus hold together? The proton and neutron in a deuteron do not attract each other and the two protons in an alpha particle repel each other electrically and yet stay bound. What glues them?

are two answers on the table.

## The Standard-Model answer: QCD

Quantum Chromodynamics — the theory of quarks and gluons — was written down in **1973**. Its residual, leftover force between colour-neutral protons and neutrons is what textbooks call the strong nuclear force, and it is the reason nuclei are supposed to exist.

QCD is a genuine triumph *at its own scale*: asymptotic freedom, the hadron spectrum, jets in colliders, deep-inelastic scattering. On those it is superb.

But on the specific job of predicting a **nuclear binding energy**, from first principles and without fitting, the record after fifty-three years is blank:

• **No parameter-free number for the deuteron.** Not the alpha. Not anything.
• **Lattice QCD** can reach light nuclei only at *unphysical* quark masses — heavy pions, hundreds of MeV too large — and the extrapolation back to the real world is uncontrolled and openly disputed between groups.
• The theories that *do* reproduce nuclei — chiral effective field theory, phenomenological potentials — are **fitted** to nuclear data first: their low-energy constants are read off the very binding energies they then "explain."

So the number that motivates the strong force is still not among the numbers the strong force predicts.

## The Coulomb answer: RealNucleus

In the RealNucleus picture there is no strong force and no weak force. A nucleus is nothing but **protons and electrons as charge clouds**, bound by the ordinary **Coulomb** attraction — the same electric law that binds atoms and molecules, read with the charges rearranged. The neutron is a bound proton–electron pair; the deuteron is **2 protons + 1 electron**, two positive charges glued by one negative one — the nuclear cousin of the molecular ion H₂⁺.

From that, with the **electric force only** and a **single scale** fixed on the deuteron — nothing else fitted — the model delivers:

• the **alpha binding energy, ~28 MeV** — the very number QCD cannot give;
• the **alpha/deuteron binding ratio, 13.1** against a measured 12.7 — a genuinely *parameter-free* prediction, because a ratio does not see the overall scale;
• the whole **alpha-conjugate ladder ⁴He … ⁴⁰Ca at ~107%**, with near-constant **binding per nucleon** (saturation) *emerging* rather than assumed;
• **D+D→⁴He fusion**, **alpha decay** (Gamow / Geiger–Nuttall), and phase-triggered beta decay, all from the same functional;
• and a proof that the **electron's mass is irrelevant** to the result — a genuinely light electron, relaxing on its own, chooses to be flat and charge-continuous, so the nuclear scale is set by the *heavy proton* and the atomic scale by the *light electron*: two sizes, one Coulomb law.

## The honest caveat

This is *one scale*, not literally zero input — the deuteron energy sets the unit. But a unit is not a fit: once it is chosen, every **ratio** and the **shape** of the binding-per-nucleon curve are predictions, not adjustments. There are real open problems too — the spin–statistics of the electron-in-nucleus, closed-shell structure, and RealNucleus stays deliberately silent on the neutrino. None of it is settled.

## The point

The alpha particle's ~28 MeV is the canonical thing the strong force was invented to account for. It is reproduced, to about 107%, with a single scale, by a model that **contains no strong force at all**.

That does not retire QCD, which remains the right theory of quarks and gluons. But it makes an uncomfortable question legitimate and, after fifty-three years, still unanswered:

**If a nucleus can be bound by the electric force alone, how much of the strong-force machinery is actually needed to explain why nuclei exist — and how much have we been assuming?**

Full argument, computations, and simulations are in the paper "RealNucleus" and at [claes542.github.io/RealMolecule](https://claes542.github.io/RealMolecule/gallery.html).*